Solve for all values of x : \[ \sqrt{x}-2=\sqrt{x-6} \]

To solve the equation \( \sqrt{x} - 2 = \sqrt{x - 6} \), first isolate one of the square root terms: 1. Add 2 to both sides: \[ \sqrt{x} = \sqrt{x - 6} + 2 \] 2. Next, square both sides to eliminate the square roots: \[ (\sqrt{x})^2 = (\sqrt{x - 6} + 2)^2 \] This simplifies to: \[ x = (x - 6) + 4\sqrt{x - 6} + 4 \] Which can be rewritten as: \[ x = x - 6 + 4\sqrt{x - 6} + 4 \] 3. Simplifying the right side gives: \[ x = x - 2 + 4\sqrt{x - 6} \] 4. Subtract \(x\) from both sides: \[ 0 = -2 + 4\sqrt{x - 6} \] 5. Add 2 to both sides: \[ 2 = 4\sqrt{x - 6} \] 6. Divide by 4: \[ \frac{1}{2} = \sqrt{x - 6} \] 7. Square both sides again: \[ \left(\frac{1}{2}\right)^2 = x - 6 \] \[ \frac{1}{4} = x - 6 \] 8. Add 6 to both sides: \[ x = 6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4} \] Now, we need to check if this solution is valid by substituting back into the original equation. For \(x = \frac{25}{4}\): \[ \sqrt{\frac{25}{4}} - 2 = \frac{5}{2} - 2 = \frac{1}{2} \] and \[ \sqrt{\frac{25}{4} - 6} = \sqrt{\frac{25}{4} - \frac{24}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \] Both sides are equal, confirming \(x = \frac{25}{4}\) is indeed a solution. Thus, the final solution is: \[ \boxed{\frac{25}{4}} \]